#495273

order	512
nilpotency class	4
isoclinism	isoclinic class of #33595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1236 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(2,4)(5,6,11,17,33,16,34,48,12,19,9,15,10,18,35,47)(7,20,24,38,23,49,32,13,26,39,22,50,25,46,36,40)(8,27,31,60,61,42,63,43,28,45,29,59,30,14,41,44)(21,52,56,65,55,51,62,58,53,67,54,37,57,66,64,68);(1,2,3,4)(5,7)(6,13)(8,21)(9,32)(10,23)(11,36)(12,26)(14,37)(15,46)(16,38)(17,49)(18,50)(19,40)(20,48)(22,34)(24,35)(25,33)(27,58)(28,53)(29,62)(30,55)(31,64)(39,47)(41,56)(42,65)(43,52)(44,67)(45,68)(51,60)(54,63)(57,61)(59,66);(5,8,10,30,12,28,33,61)(6,14,16,27,19,42,18,45)(7,21,23,55,26,53,25,57)(9,29,34,63,11,31,35,41)(13,37,38,58,40,65,50,68)(15,43,47,59,17,44,48,60)(20,51,46,52,39,66,49,67)(22,54,36,64,24,56,32,62);(8,28)(14,42)(21,53)(27,45)(29,31)(30,61)(37,65)(41,63)(43,44)(51,66)(52,67)(54,56)(55,57)(58,68)(59,60)(62,64)

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,c^2=f,e^2=hi,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fi,e^b=eh,g^b=gh,h^b=hi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402469);